Observer Consensus: From Local to Global

“Multiple local observers reach consensus through causal structure, emerging global spacetime.”

🎯 Core of This Article

In previous articles, we understood causal structure from geometric perspective. Now we answer a profound question:

Is spacetime objective or subjective?

Answer is neither purely objective nor purely subjective, but:

$$\boxed{\text{Spacetime}=\text{Consensus Geometry of Multiple Observers}}$$

Core Idea:

• Each observer can only access local causal horizon
• Different observers exchange information through communication
• Observers reach agreement through three-level consensus mechanism
• Consensus convergence forms global spacetime structure

This is from local to global paradigm!

👥 Analogy: Blind Men and Elephant

Modern version of classic story “Blind Men and Elephant”:

graph TB
    subgraph "Four Blind Men"
        O1["Blind Man 1<br/>Touches Leg"]
        O2["Blind Man 2<br/>Touches Trunk"]
        O3["Blind Man 3<br/>Touches Tail"]
        O4["Blind Man 4<br/>Touches Ear"]
    end

    subgraph "Local Cognition"
        O1 -->|"Thinks"| P1["Pillar"]
        O2 -->|"Thinks"| P2["Pipe"]
        O3 -->|"Thinks"| P3["Rope"]
        O4 -->|"Thinks"| P4["Fan"]
    end

    subgraph "Communication and Consensus"
        P1 -.->|"Exchange"| COM["Common Discussion"]
        P2 -.->|"Exchange"| COM
        P3 -.->|"Exchange"| COM
        P4 -.->|"Exchange"| COM
        COM -->|"Consensus"| ELEPHANT["Elephant"]
    end

    style O1 fill:#e1f5ff
    style O2 fill:#fff4e1
    style O3 fill:#ffe1e1
    style O4 fill:#e1ffe1
    style ELEPHANT fill:#f0f0f0,stroke:#ff6b6b,stroke-width:3px

Analogy to GLS Theory:

• Blind Men: Local observers (can only see events within causal horizon)
• Elephant Parts: Local causal diamonds
• Exchange: Communication through light signals
• Consensus: Čech consistency + state convergence + model unification
• Complete Elephant: Global spacetime manifold

Key Insight:

• No “God’s eye view” directly seeing complete spacetime
• Global spacetime emerges from consensus of local observers
• This is observer version of emergent gravity

📐 Formal Definition of Observer

Observer Nine-Tuple

In GLS theory, an observer $O_i$ is formalized as nine-tuple:

$$O_i=(C_i,{\prec }_i,{\Lambda }_i,{\mathcal{A}}_i,{\omega }_i,{\mathcal{M}}_i,U_i,u_i,\{{\mathcal{C}}_{ij}\})$$

Let’s explain one by one:

1. Causal Horizon $C_i\subset M$

Spacetime region that observer can access, usually their past light cone:

$$C_i=J^{-}(O_i)=\{p\in M\mid p\prec O_i\}$$

Physical Meaning: Observer can only know events in their past light cone.

2. Local Partial Order ${\prec }_i\subset C_i\times C_i$

Causal relation defined by observer in their horizon (satisfies reflexivity, transitivity, antisymmetry).

Recall: Local partial order gluing discussed in Article 3.

3. Event Partition ${\Lambda }_i$

Observer’s resolution or coarse-graining of events.

Examples:

• Classical observer: ${\Lambda }_i$ might be macroscopic events (“see light”)
• Quantum observer: ${\Lambda }_i$ might be set of measurement projection operators

Mathematical Structure: ${\Lambda }_i$ is a σ-algebra (measure theory) or lattice (lattice theory) on $C_i$.

4. Observable Algebra ${\mathcal{A}}_i$

Operator algebra formed by physical quantities observer can measure.

Quantum Field Theory: ${\mathcal{A}}_i$ is von Neumann algebra, generated by local field operators:

$${\mathcal{A}}_i=⟨\{\varphi (x){\}}_{x\in C_i}{⟩}^{\prime \prime }$$

(Double prime means “double commutant”, generating von Neumann algebra)

5. Quantum State ${\omega }_i$

Observer’s knowledge state of system, represented as state on observable algebra:

$${\omega }_i:{\mathcal{A}}_i\to \mathbb{C},{\omega }_i(A)=⟨A⟩$$

Satisfying:

• Linearity: ${\omega }_i(\alpha A+\beta B)=\alpha {\omega }_i(A)+\beta {\omega }_i(B)$
• Positivity: ${\omega }_i(A^{\ast }A)\ge 0$
• Normalization: ${\omega }_i(\mathbb{I})=1$

6. Physical Model ${\mathcal{M}}_i$

Observer’s prior assumptions about physical laws, including:

• Hamiltonian (or action)
• Field equations
• Symmetries

Bayesian Perspective: ${\mathcal{M}}_i$ is observer’s “theory space”.

7. Utility Function $U_i$

Observer’s objective function or preference.

Examples:

• Maximize information acquisition: $U_i=S({\rho }_i)$ (entropy maximization)
• Minimize energy: $U_i=-⟨H_i⟩$
• Maximize accuracy: $U_i=-D({\rho }_{\mathrm{true}}\Vert {\rho }_i)$ (minimize relative entropy)

Decision Theory: Observer chooses measurement strategy according to $U_i$.

8. Four-Velocity $u_i^{\mu }$

Observer’s motion state (timelike unit vector):

$$u_i^{\mu }{u}_{i\mu }=-1$$

Physical Meaning: Defines observer’s “time direction” and reference frame.

9. Communication Graph $\{{\mathcal{C}}_{ij}\}$

Communication channels between observers:

$${\mathcal{C}}_{ij}:O_i\to O_j$$

Including:

• Communication delay (light speed limit)
• Communication bandwidth
• Communication reliability (noise)

graph TB
    subgraph "Observer Nine-Tuple"
        CI["Step 1: Causal Horizon C_i"]
        PREC["Step 2: Local Partial Order ≺_i"]
        LAMBDA["Step 3: Event Partition Λ_i"]
        ALG["Step 4: Observable Algebra 𝒜_i"]
        STATE["Step 5: Quantum State ω_i"]
        MODEL["Step 6: Physical Model ℳ_i"]
        UTIL["Step 7: Utility Function U_i"]
        VEL["Step 8: Four-Velocity u_i"]
        COM["Step 9: Communication Graph {𝒞_ij}"]
    end

    CI --> GEO["Geometric Information"]
    PREC --> GEO
    LAMBDA --> GEO

    ALG --> QM["Quantum Information"]
    STATE --> QM
    MODEL --> QM

    UTIL --> DEC["Decision Information"]
    VEL --> DEC

    COM --> NET["Network Information"]

    style CI fill:#e1f5ff
    style STATE fill:#fff4e1
    style MODEL fill:#ffe1e1
    style COM fill:#e1ffe1

🔗 Three-Level Consensus Mechanism

Observers reach agreement through three levels of consensus:

Level 1: Causal Consensus

Goal: Different observers agree on causal relations.

Mechanism: Čech consistency condition (detailed in Article 3)

$${\prec }_i{|}_{C_i\cap C_j}={\prec }_j{|}_{C_i\cap C_j}$$

Convergence Criterion:

• All observers agree on causal order in overlapping regions
• Local partial orders can be glued into global partial order

graph LR
    O1["Observer O₁<br/>Partial Order ≺₁"] -->|"Overlapping Region"| OVERLAP["C₁ ∩ C₂"]
    O2["Observer O₂<br/>Partial Order ≺₂"] -->|"Overlapping Region"| OVERLAP

    OVERLAP -->|"Čech Consistency"| CHECK["≺₁ = ≺₂ ?"]
    CHECK -->|"Yes"| CONSENSUS["Causal Consensus Reached"]
    CHECK -->|"No"| UPDATE["Update Partial Order"]
    UPDATE --> O1
    UPDATE --> O2

    style O1 fill:#e1f5ff
    style O2 fill:#fff4e1
    style CONSENSUS fill:#e1ffe1

Level 2: State Consensus

Goal: Quantum states of different observers converge in common region.

Mechanism: Relative entropy Lyapunov function

Define consensus state ${\omega }_{\ast }\in \mathcal{S}({\mathcal{A}}_{\mathrm{com}})$ (on common observable algebra ${\mathcal{A}}_{\mathrm{com}}={\bigcap }_i{\mathcal{A}}_i$).

Lyapunov Function:

$${\Phi }^{(t)}:=\sum _i{\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast })$$

where:

• $D({\omega }_i\Vert {\omega }_{\ast })=\mathrm{tr}({\omega }_i\log {\omega }_i)-\mathrm{tr}({\omega }_i\log {\omega }_{\ast })$: Relative entropy
• ${\lambda }_i\ge 0$: Weights (${\sum }_i{\lambda }_i=1$)

Convergence Theorem:

Under reasonable assumptions (communication, measurement, update), ${\Phi }^{(t)}$ monotonically decreases:

$$\frac{\mathrm{d}{\Phi }^{(t)}}{\mathrm{d}t}\le 0$$

Finally:

$${\omega }_i^{(t\to \infty )}\to {\omega }_{\ast }$$

Physical Meaning:

• Relative entropy measures “difference” between states
• Lyapunov function decreasing → observers’ cognition converges
• Finally reach consensus state

graph TB
    subgraph "Initial State (t=0)"
        W1["ω₁⁽⁰⁾"] -.->|"Large Difference"| W2["ω₂⁽⁰⁾"]
        W2 -.->|"Large Difference"| W3["ω₃⁽⁰⁾"]
    end

    subgraph "Evolution (t > 0)"
        W1 -->|"Communication+Update"| W1T["ω₁⁽ᵗ⁾"]
        W2 -->|"Communication+Update"| W2T["ω₂⁽ᵗ⁾"]
        W3 -->|"Communication+Update"| W3T["ω₃⁽ᵗ⁾"]
    end

    subgraph "Consensus (t→∞)"
        W1T -->|"Converge"| WSTAR["ω* (Consensus State)"]
        W2T -->|"Converge"| WSTAR
        W3T -->|"Converge"| WSTAR
    end

    LYAP["Lyapunov Function<br/>Φ⁽ᵗ⁾ Monotonically Decreasing"] -.-> W1T
    LYAP -.-> W2T
    LYAP -.-> W3T

    style WSTAR fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px
    style LYAP fill:#fff4e1

Level 3: Model Consensus

Goal: Physical models (theories) of different observers converge.

Mechanism: Bayesian update + large deviation theory

Each observer maintains a model distribution ${\mathbb{P}}_i^{(t)}$ (on model space $\mathcal{M}$).

Bayesian Update:

$${\mathbb{P}}_i^{(t+1)}(\mathcal{M}|\text{data})\propto {\mathbb{P}}_i^{(t)}(\mathcal{M})\cdot \mathcal{L}(\text{data}|\mathcal{M})$$

where $\mathcal{L}$ is likelihood function.

Convergence Criterion (Donsker-Varadhan large deviation principle):

$$D_{\mathrm{KL}}({\mathbb{P}}_i^{(t)}\Vert {\mathbb{P}}_{\ast })\to 0\text{as}t\to \infty$$

where ${\mathbb{P}}_{\ast }$ is consensus model distribution, determined by true statistics of data.

Physical Meaning:

• Observers update theory through experimental data
• Different observers eventually converge to same theory
• This is mathematicalization of scientific consensus formation

graph LR
    DATA["Observation Data"] -->|"Bayesian Update"| P1["Observer 1's Model Distribution<br/>ℙ₁⁽ᵗ⁾"]
    DATA -->|"Bayesian Update"| P2["Observer 2's Model Distribution<br/>ℙ₂⁽ᵗ⁾"]
    DATA -->|"Bayesian Update"| P3["Observer 3's Model Distribution<br/>ℙ₃⁽ᵗ⁾"]

    P1 -->|"KL Divergence Decreasing"| PSTAR["Consensus Model Distribution<br/>ℙ*"]
    P2 -->|"KL Divergence Decreasing"| PSTAR
    P3 -->|"KL Divergence Decreasing"| PSTAR

    PSTAR -.->|"Corresponds to"| THEORY["Optimal Physical Theory"]

    style DATA fill:#e1f5ff
    style PSTAR fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px
    style THEORY fill:#fff4e1

🌐 Communication Graph and Information Propagation

Communication Graph Structure

Communication of observer network $\mathcal{N}=\{O_1,\dots ,O_N\}$ is described by graph $G=(V,E)$:

• Vertices $V=\{O_1,\dots ,O_N\}$: Observers
• Edges $E=\{{\mathcal{C}}_{ij}\}$: Communication channels

Adjacency Matrix $W=(w_{ij})$:

$$w_{ij}=\{\begin{array}{ll}>0& \text{if}{O}_i\text{can communicate with}{O}_j\\ 0& \text{otherwise}\end{array}$$

Physical Constraints:

• Causality: Only when $C_i\cap C_j\ne \varnothing$, $w_{ij}>0$
• Symmetry (optional): $w_{ij}=w_{ji}$ (bidirectional communication)
• Light Speed Limit: Communication delay $\ge$ light propagation time

graph TB
    subgraph "Observer Network"
        O1["O₁"] ---|"w₁₂"| O2["O₂"]
        O2 ---|"w₂₃"| O3["O₃"]
        O3 ---|"w₃₄"| O4["O₄"]
        O1 ---|"w₁₄"| O4
        O2 ---|"w₂₄"| O4
    end

    subgraph "Communication Matrix W"
        W["[w_ij]<br/>Weight Matrix"]
    end

    O1 -.-> W
    O2 -.-> W
    O3 -.-> W
    O4 -.-> W

    style O1 fill:#e1f5ff
    style O2 fill:#fff4e1
    style O3 fill:#ffe1e1
    style O4 fill:#e1ffe1

Consensus Algorithm

Discrete Time Update (e.g., linear consensus algorithm):

$${\omega }_i^{(t+1)}=(1-ϵ){\omega }_i^{(t)}+ϵ\sum _j{w}_{ij}{\omega }_j^{(t)}$$

where:

• $ϵ\in (0,1)$: Learning rate
• $w_{ij}$: Information weight from observer $j$ (normalized: ${\sum }_j{w}_{ij}=1$)

Convergence Condition (graph theory):

• Graph $G$ is connected (any two observers can be connected by path)
• Weight matrix $W$ is doubly stochastic (row and column sums are 1)

Convergence Rate: Controlled by second largest eigenvalue ${\lambda }_2(W)$ of $W$:

$$\Vert {\omega }_i^{(t)}-{\omega }_{\ast }\Vert \le C\cdot {\lambda }_2(W{)}^t$$

(${\lambda }_2<1$, smaller means faster convergence)

📊 Emergence from Local to Global

Construction of Global Spacetime

Step 1: Local observers define local causal diamonds $\{D_i\}$

Each observer $O_i$ defines causal diamond family in their horizon $C_i$.

Step 2: Causal consensus forms global partial order

Through Čech consistency, local partial orders glue into global partial order $(M,\prec )$.

Step 3: State consensus determines global state

Through relative entropy convergence, local states $\{{\omega }_i\}$ converge to consensus state ${\omega }_{\ast }$.

Step 4: Model consensus determines physical laws

Through Bayesian update, local models $\{{\mathcal{M}}_i\}$ converge to consensus theory.

Step 5: Emergence of Metric

From consensus partial order $\prec$ and consensus state ${\omega }_{\ast }$, can reconstruct spacetime metric $g_{\mu \nu }$:

$$g_{\mu \nu }=\text{function}(\prec ,{\omega }_{\ast },{\mathcal{M}}_{\ast })$$

Specific construction:

• Partial order $\prec$ → light cone structure → conformal class $[g]$
• State ${\omega }_{\ast }$ → energy-momentum tensor $T_{\mu \nu }$ (through modular Hamiltonian)
• Einstein equation → complete metric $g_{\mu \nu }$

graph TB
    subgraph "Local Observers"
        O1["O₁: (C₁, ≺₁, ω₁)"]
        O2["O₂: (C₂, ≺₂, ω₂)"]
        O3["O₃: (C₃, ≺₃, ω₃)"]
    end

    subgraph "Consensus Layer"
        CAUSAL["Causal Consensus<br/>≺"]
        STATE["State Consensus<br/>ω*"]
        MODEL["Model Consensus<br/>ℳ*"]
    end

    subgraph "Global Spacetime"
        METRIC["Metric g_μν"]
        SPACETIME["Spacetime (M, g)"]
    end

    O1 -->|"Čech"| CAUSAL
    O2 -->|"Čech"| CAUSAL
    O3 -->|"Čech"| CAUSAL

    O1 -->|"Relative Entropy"| STATE
    O2 -->|"Relative Entropy"| STATE
    O3 -->|"Relative Entropy"| STATE

    O1 -->|"Bayesian"| MODEL
    O2 -->|"Bayesian"| MODEL
    O3 -->|"Bayesian"| MODEL

    CAUSAL -->|"Light Cone"| METRIC
    STATE -->|"T_μν"| METRIC
    MODEL -->|"Einstein Equation"| METRIC

    METRIC --> SPACETIME

    style O1 fill:#e1f5ff
    style O2 fill:#fff4e1
    style O3 fill:#ffe1e1
    style SPACETIME fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px

Observer Interpretation of Emergent Gravity

Traditional View: Spacetime metric $g_{\mu \nu }$ is fundamental, observers move in it.

GLS View: Spacetime metric emerges from observer consensus:

$$\text{Observer Network}\stackrel{\text{Causal+State+Model Consensus}}{\to }\text{Spacetime Metric}$$

Analogy:

• Temperature (macroscopic) ← Molecular Motion (microscopic)
• Spacetime Metric (macroscopic) ← Observer Consensus (microscopic)

This is concrete realization of emergent gravity!

🔍 Example: GPS Satellite Network

Scenario

Around Earth there are $N\sim 30$ GPS satellites, each is an observer:

$$O_i=(\text{orbit},\text{atomic clock},\text{signal receiver},\dots )$$

Local Horizon

Each satellite can only see:

• Its own past light cone $C_i=J^{-}(O_i)$
• Signals received from other satellites

Communication Graph

$$w_{ij}=\{\begin{array}{ll}>0& \text{if satellites}i\text{and}j\text{are visible}\\ 0& \text{otherwise}\end{array}$$

Causal Consensus

Satellites confirm event order through signal exchange:

• Did event A occur before event B?
• Need to correct relativistic effects (gravitational time dilation, motion time dilation)

State Consensus

Satellites synchronize time:

• Each satellite has atomic clock (local time $t_i$)
• Synchronize to GPS time $t_{\mathrm{GPS}}$ through consensus algorithm

Lyapunov Function:

$${\Phi }^{(t)}=\sum _i(t_i^{(t)}-t_{\mathrm{GPS}}{)}^2$$

Decreasing to zero → time synchronization complete!

Model Consensus

All satellites use same theory:

• General relativity
• Schwarzschild metric (Earth’s gravitational field)

If some satellite’s model is wrong (e.g., forgets general relativistic correction), its predictions will systematically deviate, eventually “corrected” by other satellites (through Bayesian update).

GPS is daily realization of GLS theory!

graph TB
    subgraph "GPS Satellite Network"
        SAT1["Satellite 1<br/>Atomic Clock t₁"]
        SAT2["Satellite 2<br/>Atomic Clock t₂"]
        SAT3["Satellite 3<br/>Atomic Clock t₃"]
        SAT4["Satellite 4<br/>Atomic Clock t₄"]
    end

    subgraph "Consensus Mechanism"
        SYNC["Time Synchronization<br/>t_GPS"]
        GR["General Relativistic Correction"]
    end

    SAT1 -.->|"Communication"| SYNC
    SAT2 -.->|"Communication"| SYNC
    SAT3 -.->|"Communication"| SYNC
    SAT4 -.->|"Communication"| SYNC

    SYNC -->|"Correction"| GR
    GR -.->|"Feedback"| SAT1
    GR -.->|"Feedback"| SAT2
    GR -.->|"Feedback"| SAT3
    GR -.->|"Feedback"| SAT4

    style SAT1 fill:#e1f5ff
    style SAT2 fill:#fff4e1
    style SAT3 fill:#ffe1e1
    style SAT4 fill:#e1ffe1
    style SYNC fill:#f0f0f0,stroke:#ff6b6b,stroke-width:3px

💡 Key Points Summary

1. Observer Nine-Tuple

$$O_i=(C_i,{\prec }_i,{\Lambda }_i,{\mathcal{A}}_i,{\omega }_i,{\mathcal{M}}_i,U_i,u_i,\{{\mathcal{C}}_{ij}\})$$

Contains causal, quantum, model, decision, motion, communication information.

2. Three-Level Consensus

• Causal Consensus: Čech consistency, ${\prec }_i={\prec }_j$ in overlapping regions
• State Consensus: Relative entropy Lyapunov function, ${\omega }_i\to {\omega }_{\ast }$
• Model Consensus: Bayesian update, ${\mathbb{P}}_i\to {\mathbb{P}}_{\ast }$

3. Relative Entropy Lyapunov Function

$${\Phi }^{(t)}=\sum _i{\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast }),\frac{\mathrm{d}\Phi }{\mathrm{d}t}\le 0$$

Ensures consensus convergence.

4. Communication Graph

Weight matrix $W=(w_{ij})$, connectivity ensures consensus.

5. Emergent Spacetime

$$\text{Observer Consensus}⟹\text{Spacetime Metric}$$

Global spacetime emerges from consensus of local observers.

🤔 Thought Questions

Question 1: What If Observers Cannot Communicate ($w_{ij}=0$ for all $i\ne j$)?

Hint: Consider graph connectivity.

Answer: Cannot reach consensus! Each observer has their own “subjective spacetime”, cannot form unified global geometry. This is similar to multiple causally isolated universe islands. Physically, this corresponds to regions beyond cosmological horizon: cannot communicate, therefore cannot verify causal consistency.

Question 2: How Should Weights ${\lambda }_i$ of Lyapunov Function Be Chosen?

Hint: Consider observer’s “credibility” or “measurement precision”.

Answer: ${\lambda }_i$ should reflect observer’s information quality. For example:

• Observers with high measurement precision → larger ${\lambda }_i$
• Observers with more observation data → larger ${\lambda }_i$
• Can dynamically adjust: ${\lambda }_i^{(t)}\propto (\text{inverse of information entropy}{)}^{-1}$

This is similar to weighted average, more reliable observers have greater influence.

Question 3: What If Some Observer Is “Malicious” (Provides Wrong Information)?

Hint: Consider Byzantine generals problem.

Answer: This is Byzantine consensus problem! In quantum information, if:

• Number of malicious observers $\le N/3$ ($N$ is total number)
• Other observers can verify information consistency (e.g., through Markov property test)

then consensus can still be reached, but requires more complex algorithms (e.g., Byzantine fault-tolerant consensus).

In GLS theory, causal consistency (Čech condition) provides natural verification mechanism!

Question 4: What Is Connection Between Observer Consensus and AdS/CFT?

Hint: Consider multiple “observer regions” of boundary CFT.

Answer: In AdS/CFT:

• Bulk AdS: Global spacetime
• Boundary CFT: Can be divided into multiple subregions (corresponding to different “observers”)
• Entanglement Wedges: Causal horizon of each boundary region
• Consensus: Entanglement structure of subregions must be consistent to reconstruct complete bulk

GLS theory of observer consensus can be seen as observer interpretation of AdS/CFT!

📖 Source Theory References

Content of this article mainly from following source theories:

Core Source Theory

Document: docs/euler-gls-causal/observer-properties-consensus-geometry-causal-network.md

Key Content:

• Nine-tuple formalization of observers
• Čech-type consistency conditions (causal consensus)
• Relative entropy Lyapunov function (state consensus)
• Bayesian update and large deviation (model consensus)
• Communication graph and information propagation
• Spacetime emergence from local to global

Important Theorem (original text):

“Observers reach agreement through three-level consensus (causal, state, model), forming global spacetime structure. Relative entropy Lyapunov function ensures consensus convergence.”

Related Literature

Quantum Network Consensus:

• Olfati-Saber & Murray (2004): Classical consensus algorithms
• Boyd et al. (2006): Distributed optimization
• Quantum generalization: Consensus of quantum states (recent research)

Bayesian Inference:

• Jaynes (1957): Maximum entropy principle
• MacKay (2003): Bayesian machine learning

Byzantine Consensus:

• Lamport et al. (1982): Byzantine generals problem
• Blockchain: Decentralized consensus (Bitcoin, Ethereum)

🎯 Next Steps

We’ve completed core content of Causal Structure chapter! Next article is summary of this chapter, connecting all concepts to form complete picture.

Next Article: 07-causal-summary_en.md - Complete Picture of Causal Structure

There, we will see:

• Review of seven articles on causal structure
• Complete connection of trinity (geometry-time-entropy)
• Relationship with previous chapters (boundary theory, unified time)
• Leading to next chapter (topological constraints)
• Core position of causal structure in GLS theory

Back: Causal Structure Chapter Overview

Previous: 05-markov-property_en.md